On spontaneous breaking of conformal symmetry by probe flavour D-branes
Omer Ben-Ami
1
Stanislav Kuperstein
0
Jacob Sonnenschein
1
0
Institut de Physique Theorique
, CEA Saclay, CNRS URA 2306, F-91191 Gif-sur-Yvette,
France
1
Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel-Aviv University
, Ramat-Aviv 69978,
Israel
We explore the possibilities of breaking conformal symmetry spontaneously by introducing flavour branes into conformal holographic backgrounds in the probe limit. A prototype model of such a mechanism is based on placing D7-D 7 configuration in the Klebanov-Witten conifold based model. In this paper we generalize this model. We conjecture on the required topology of the backgrounds and the corresponding probe brane embeddings. We identify several models that obey these requirements and admit spontaneous breaking of conformal invariance. These include type IIB conifold based examples, dual to defect field theories based on the conifold, and type IIA constructions based on the ABJM model. We identify the dilaton, the corresponding Goldstone boson, discuss its effective action and address the a-term. We briefly discuss the relevance of these models to the pseudo dilaton.
1 Introduction
Mass and VEV deformation by probe D7-Branes
More U-shape examples
3.1 D5 probe wrapping AdS3 S3 in AdS5 T 1,1
3.2 D5 probe wrapping AdS4 S2 in AdS5 T 1,1
3.3 D3 probe wrapping AdS2 S2 in AdS5 T 1,1
3.4 D4 probe wrapping AdS3 CP1 in AdS4 CP3
4 The dilaton
Models with a pseudo-dilaton
6 Summary and open questions
1 Introduction
Known examples of spontaneous conformal symmetry breaking are scarce. Although
classically it may not be hard to find conformal invariant (interacting) field theories, it becomes
a highly non-trivial task at the quantum level. This is due to the fact that one has to
introduce a scale in order to regularize the theory. This scale can explicitly break the
symmetry by telling us that marginal operators are not exactly marginal (i.e. QCDs
function) or, rather miraculously, tell us that they are (the perturbative Banks-Zaks [1]
interacting fixed point).
Finding such conformal theories is the first step. The second is to break the symmetry
spontaneously. This means that the deformation of the interacting conformal field theory,
initiating an RG-flow, should be a VEV deformation. VEV deformations with flat
directions are especially hard to come by in conformal theories where Mexican hat models are,
of course, out of the question. They do come by, however, in supersymmetric conformal
theories where flat directions are present also at the quantum level. The prime example
being N = 4 SUSY with an RG flow from SU(N ) to SU(N 1), initiated by giving a VEV
to one of the scalars. Integrating out the massive fields, the theory flows from one fixed
point to the other. Although not necessarily perturbative, in these models only operators
that already appear in the Lagrangian acquire a VEV. There is another well known
possibility of breaking a symmetry spontaneously by strong coupling effects. This is usually
also referred to as dynamical symmetry breaking. This is what happens in nature in chiral
symmetry breaking where the quark condensate is the order parameter for the symmetry
breaking. Strongly coupled conformal theories can in principle display the same behavior,
thus breaking conformal symmetry spontaneously.
Generally speaking, there is no reason why there should be a flat potential for the
scalar(s) in the theory (whether they are perturbative and already appear in the UV
Lagrangian or a mesonic like operator). Supersymmetry, however, provides a mechanism
which preservers the flat directions and allows for the scalar to receive an arbitrary VEV
thus breaking the symmetry spontaneously. The models we explore break supersymmetry
but still allow for the conformal symmetry to be broken spontaneously. We suspect that
this is an artifact of our probe approximation and all corrections to the potential should
appear once backreaction is included.
Our main motivation is to search for holographic models that admit spontaneous
breaking of conformal invariance in a certain type of holographic models.
In general, spontaneous breaking of conformal symmetry in a holographic gravity setup
can be done either by analysing holographic gravitational background with (or without)
additional bulk fields or by embedding Nf flavour D-branes in such backgrounds. Examples
of the former approach can be found in [2, 3] where domain wall geometries in arbitrary
number of dimensions interpolating between two AdS spaces, were analysed. In this paper
we follow solely the latter, D-brane embedding approach. Moreover, we will consider only
the probe limit, where one can ignore the backreaction of the embedded D-branes on the
geometry. This is easily achieved by taking the Nf Nc limit and in our discussion we will
restrict ourself only to the Nf = 1 case. We use the terminology flavour branes for branes
which reach the UV boundary (unlike g (...truncated)